Finite Group Databasebeta

← #16930233

Subgroups — #16930233

Order 4352.

566 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531694C211yesmaximal, characteristic
2#14527348C211yesmaximal, characteristic, complemented
3#14527283C211yesmaximal, characteristic, complemented
4#14527175C211yesmaximal, characteristic
5#14526685C211yesmaximal, characteristic, complemented
6#14526644C211yesmaximal, characteristic, complemented
7C17×#3077C211yesmaximal, characteristic
8C2×D4×Dic17—21no—
9#11762862C2211yescharacteristic
10#11762862C2211yescharacteristic
11#11762708—21no—
12#11762684—22no—
13#11762675C2211yescharacteristic
14#11762669—21no—
15C4×C4⋊Dic17C2211yescharacteristic
16C2×#10587502—21no—
17#11762546—21no—
18#11762546—21no—
19#11762546—22no—
20D4⋊#3622—22no—
21C17×#533C2211yescharacteristic
22C2×C4×D4×C17C2211yescharacteristic
23C17×#478—22no—
24C17×#476C412yescomplemented
25C17×#474C412yescomplemented
26C17×#428—21no—
27C17×#423C2211yescharacteristic
28C17×#418—21no—
29C22×D4×C17D411yescharacteristic
30C23×C4×C17—21no—
31C23×Dic17—22no—
32C2×M16×C17—21no—
33C17×C4⋊Q8C2×C411yescharacteristic, complemented
34C17×C4⋊Q8C2×C411yescharacteristic, complemented
35C17×#125—21no—
36C17×C22⋊Q8—21no—
37C17×C22⋊Q8—21no—
38C4×Q8×C17—21no—
39C4×Q8×C17—21no—
40C4×D4×C17—21no—
41C4×D4×C17—21no—
42C4×D4×C17C2×C412yes—
43C4×D4×C17D412yes—
44C17×#117—21no—
45C2×C4⋊C4×C17D411yescharacteristic
46C2×C22⋊C4×C17—21no—
47C2×C42×C17C2311yescharacteristic
48C2×C22⋊Dic17—22no—
49D4×Dic17—42no—
50D4×Dic17—42no—
51C2×C4⋊Dic17—22no—
52C2×C4⋊Dic17—21no—
53C2×C4⋊Dic17—21no—
54C2×C4×Dic17—21no—
55C17×#107—22no—
56C17×#106—22no—
57C17×C4⋊C8—21no—
58C17×C4⋊C8C2×C412yescomplemented
59C17×#102—22no—
60C17×#102—22no—
61#10587502—22no—
62#10587502—21no—
63#10587502D411yescharacteristic
64#10587502D411yescharacteristic
65#10587502D412yes—
66C22⋊#3622—21no—
67C22⋊#3622—21no—
68C4⋊#3622—21no—
69C4⋊#3622—21no—
70C4×#3622—21no—
71C4×#3622—21no—
72C24×C17—21no—
73C2×Q8×C17—21no—
74C2×Q8×C17—21no—
75C2×D4×C17—21no—
76C2×D4×C17—21no—
77C2×D4×C17—21no—
78C2×D4×C17C22⋊C412yes—
79C2×D4×C17SD1612yes—
80C2×D4×C17Q1612yes—
81C22×C4×C17—21no—
82C22×C4×C17—22no—
83C22×C4×C17—22no—
84C22×C4×C17—82no—
85C22×C4×C17C2×D411yescharacteristic
86C22×C4×C17C2×D411yescharacteristic
87C22×C4×C17C2×D411yescharacteristic
88C22×Dic17—21no—
89C22×Dic17—22no—
90C22×Dic17—21no—
91C22×Dic17—44no—
92C22×Dic17—44no—
93C22×Dic17—82no—
94M16×C17—41no—
95M16×C17—41no—
96C2×C8×C17—22no—
97C4⋊C4×C17—21no—
98C4⋊C4×C17—21no—
99C4⋊C4×C17—21no—
100C4⋊C4×C17—21no—
101C4⋊C4×C17—21no—
102C4⋊C4×C17—41no—
103C4⋊C4×C17—41no—
104C4⋊C4×C17C22⋊C412yes—
105C22⋊C4×C17—22no—
106C22⋊C4×C17—22no—
107C22⋊C4×C17—41no—
108C42×C17—41no—
109C42×C17C22×C411yescharacteristic
110C42×C17C2×D411yescharacteristic
111C42×C17#4111yescharacteristic
112C42×C17#4111yescharacteristic
113C22⋊Dic17—44no—
114C4⋊Dic17—42no—
115C4⋊Dic17—41no—
116C4⋊Dic17—41no—
117C4⋊Dic17—82no—
118C4×Dic17—41no—
119C4×Dic17—41no—
120C2×#3622—41no—
121C2×#3622—41no—
122#9175—171nomaximal, Sylow
123C23×C17—22no—
124C23×C17—22no—
125C23×C17—22no—
126C23×C17—42no—
127C23×C17—42no—
128C23×C17#12011yescharacteristic
129Q8×C17—81no—
130Q8×C17—81no—
131D4×C17—22no—
132D4×C17—22no—
133D4×C17—22no—
134D4×C17#10314yes—
135C2×C4×C17—22no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—42no—
139C2×C4×C17—41no—
140C2×C4×C17—41no—
141C2×C4×C17—82no—
142C2×C4×C17—82no—
143C2×C4×C17C2×C22⋊C411yescharacteristic
144C2×C4×C17C4×D411yescharacteristic
145C2×C4×C17C4×D411yescharacteristic
146C2×C4×C17#12111yescharacteristic
147C2×C4×C17#12111yescharacteristic
148C2×C4×C17#12311yescharacteristic
149C2×C4×C17C2×SD1611yescharacteristic
150C2×C4×C17C2×Q1611yescharacteristic
151C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C17#13711yescharacteristic
153C2×Dic17—22no—
154C2×Dic17—41no—
155C2×Dic17—41no—
156C2×Dic17—42no—
157C2×Dic17—42no—
158C2×Dic17—41no—
159C2×Dic17—41no—
160C2×Dic17—41no—
161C2×Dic17—42no—
162C2×Dic17—88no—
163C8×C17—42no—
164#3622—81no—
165#3622—81no—
166#3077D1711yescharacteristic
167#2349—171no—
168#1518—171no—
169#1503—171no—
170#1478—171no—
171#1303—171no—
172#1286—171no—
173C22×C17—21no—
174C22×C17—21no—
175C22×C17—24no—
176C22×C17—44no—
177C22×C17—44no—
178C22×C17—42no—
179C22×C17—42no—
180C22×C17#38611yescharacteristic
181C22×C17#45011yescharacteristic
182C22×C17#45111yescharacteristic
183C4×C17—21no—
184C4×C17—21no—
185C4×C17—21no—
186C4×C17—21no—
187C4×C17—42no—
188C4×C17—81no—
189C4×C17—81no—
190C4×C17C2×#10311yescharacteristic
191C4×C17#41711yescharacteristic
192C4×C17#48311yescharacteristic
193C4×C17#48411yescharacteristic
194Dic17—42no—
195Dic17—81no—
196Dic17—82no—
197#533C2×D1711yescharacteristic
198C2×C4×D4—341no—
199C2×C4×D4C2×D1711yescharacteristic
200#478—22no—
201#476Dic1712yescomplemented
202#474Dic1712yescomplemented
203#431—171no—
204#431—171no—
205#428—21no—
206#423C2×D1711yescharacteristic
207#418—21no—
208#389—341no—
209#385—342no—
210#382—171no—
211#380—341no—
212C4×C4⋊C4—171no—
213C2×#95—341no—
214#328—341no—
215#328—341no—
216#328—342no—
217D4⋊C8—342no—
218C2×C17—21no—
219C2×C17—21no—
220C2×C17—44no—
221C2×C17#187211yescharacteristic
222C2×C17#190411yescharacteristic
223C2×C17#190611yescharacteristic
224C22×D4C22⋊D1711yescharacteristic
225C23×C4—21no—
226C23×C4—342no—
227C2×M16—21no—
228C4⋊Q8C2×Dic1711yescharacteristic, complemented
229C4⋊Q8C2×Dic1711yescharacteristic, complemented
230#125—21no—
231C22⋊Q8—21no—
232C22⋊Q8—21no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—682no—
238C4×D4—682no—
239C4×D4C2×Dic1712yes—
240C4×D4C22⋊D1712yes—
241#117—21no—
242C2×C4⋊C4—341no—
243C2×C4⋊C4—341no—
244C2×C4⋊C4—342no—
245C2×C4⋊C4C22⋊D1711yescharacteristic
246C2×C22⋊C4—21no—
247C2×C22⋊C4—342no—
248C2×C42—341no—
249C2×C42C22×D1711yescharacteristic
250#107—22no—
251#106—22no—
252C4⋊C8—21no—
253C4⋊C8—341no—
254C4⋊C8—341no—
255C4⋊C8C2×Dic1712yescomplemented
256#102—22no—
257#102—22no—
258C22⋊C8—341no—
259C22⋊C8—341no—
260C4×C8—341no—
261C4×C8—341no—
262#95—171no—
263#95—171no—
264#95—172no—
265#95—342no—
266#95—341no—
267C17#917511yescharacteristic, complemented, Sylow
268C24—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4#6323012yes—
275C2×D4#6323112yes—
276C2×D4C22⋊Dic1712yes—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—82no—
281C22×C4—341no—
282C22×C4—342no—
283C22×C4—341no—
284C22×C4—684no—
285C22×C4—684no—
286C22×C4—1362no—
287C22×C4D4×D1711yescharacteristic
288C22×C4D4×D1711yescharacteristic
289C22×C4C2×C22⋊D1711yescharacteristic
290M16—41no—
291M16—41no—
292C2×C8—22no—
293C2×C8—681no—
294C2×C8—681no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—682no—
303C4⋊C4—681no—
304C4⋊C4—681no—
305C4⋊C4—1362no—
306C4⋊C4C22⋊Dic1712yes—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—41no—
310C22⋊C4—684no—
311C42—41no—
312C42—681no—
313C42—681no—
314C42#6325411yescharacteristic
315C42#6325411yescharacteristic
316C42C22×Dic1711yescharacteristic
317C42C2×C22⋊D1711yescharacteristic
318C23—22no—
319C23—22no—
320C23—22no—
321C23—42no—
322C23—42no—
323C23#1058762311yescharacteristic
324Q8—81no—
325Q8—81no—
326D4—22no—
327D4—22no—
328D4—22no—
329D4#1058750614yes—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—342no—
339C2×C4—681no—
340C2×C4—681no—
341C2×C4—681no—
342C2×C4—681no—
343C2×C4—682no—
344C2×C4—682no—
345C2×C4—681no—
346C2×C4—682no—
347C2×C4—1368no—
348C2×C4D4×Dic1711yescharacteristic
349C2×C4D4×Dic1711yescharacteristic
350C2×C4#1058762111yescharacteristic
351C2×C4#1058762511yescharacteristic
352C2×C4#1058762511yescharacteristic
353C2×C4C2×#6323011yescharacteristic
354C2×C4C2×#6323111yescharacteristic
355C2×C4#1058763511yescharacteristic
356C2×C4#1058763711yescharacteristic
357C2×C4C2×C22⋊Dic1711yescharacteristic
358C8—42no—
359C8—1361no—
360C8—1361no—
361C22—21no—
362C22—21no—
363C22—24no—
364C22—44no—
365C22—44no—
366C22—42no—
367C22—42no—
368C22#1176268911yescharacteristic
369C22#1176295911yescharacteristic
370C22#1176296411yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—81no—
377C4—81no—
378C4—682no—
379C4—1362no—
380C4—1361no—
381C4C2×#1058750611yescharacteristic
382C4#1176281011yescharacteristic
383C4#1176313711yescharacteristic
384C4#1176314211yescharacteristic
385C2—21no—
386C2—21no—
387C2—44no—
388C2#1452909211yescharacteristic
389C2#1452927311yescharacteristic
390C2#1452928711yescharacteristic
subgroupG/H
centerC22#11762689
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#3077C2
socleC22×C17#386
prime radicalC22×D4×C17D4
cosocleC2×C42×C17C23
hypercenter#3077D17
hypocenterC17#9175
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #14527348C2semidirect
3. #14527283C2semidirect
5. #14526685C2semidirect
6. #14526644C2semidirect
24. C17×#476C4semidirect
25. C17×#474C4semidirect
33. C17×C4⋊Q8C2×C4semidirect
34. C17×C4⋊Q8C2×C4semidirect
58. C17×C4⋊C8C2×C4semidirect
201. #476Dic17semidirect
202. #474Dic17semidirect
228. C4⋊Q8C2×Dic17semidirect
229. C4⋊Q8C2×Dic17semidirect
255. C4⋊C8C2×Dic17semidirect
267. C17#9175semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#145316942
#145273482
#145272832
#145271752
#145266852
#145266442
C17×#30772
#917517